Which one of the following will have minimum change in its value if 5 is added to both numerator and the denominator of the fractions $\frac{2}{3},\frac{3}{4},\frac{4}{5}$and $\frac{5}{6}$?

Correct Option

The initial fraction is $\frac{5}{6}$. Its decimal equivalent is approximately 0.8333.

When 5 is added to both the numerator and the denominator, the new fraction becomes $\frac{5+5}{6+5}=\frac{10}{11}$. The decimal equivalent of $\frac{10}{11}$ is approximately 0.9091.

The change in the value of the fraction is $|0.9091-0.8333|\approx 0.0758$.

This change is the minimum among all the given options. Generally, for a proper fraction $\frac{a}{b}$ (where ), adding a positive constant to both the numerator and denominator increases its value. The closer the original fraction is to 1, the smaller the magnitude of this increase.

Incorrect Options

The changes for the other fractions are calculated as follows:

• For $\frac{2}{3}$: Initial value: $\frac{2}{3}\approx 0.6667$. After adding 5: $\frac{2+5}{3+5}=\frac{7}{8}=0.8750$. Change in value: $|0.8750-0.6667|\approx 0.2083$. This is greater than 0.0758.
• For $\frac{3}{4}$: Initial value: $\frac{3}{4}=0.7500$. After adding 5: $\frac{3+5}{4+5}=\frac{8}{9}\approx 0.8889$. Change in value: $|0.8889-0.7500|\approx 0.1389$. This is greater than 0.0758.
• For $\frac{4}{5}$: Initial value: $\frac{4}{5}=0.8000$. After adding 5: $\frac{4+5}{5+5}=\frac{9}{10}=0.9000$. Change in value: $|0.9000-0.8000|=0.1000$. This is greater than 0.0758.

Comparing the changes (0.2083, 0.1389, 0.1000, 0.0758), the minimum change occurs for the fraction $\frac{5}{6}$.